Related papers: Laurent polynomials, Eulerian numbers, and Bernste…

Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels posed two questions: Do the constant terms of a generic Laurent polynomial…

Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $\mu$ is a zero of $X_n$ if and only if the triple $(\mu,\mu+m,\mu+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the…

Let $f$ and $F$ be two polynomials satisfying $F(x)=u(x)f(x)+v(x)f'(x)$. We characterize the relation between the location and multiplicity of the real zeros of $f$ and $F$, which generalizes and unifies many known results, including the…

We generalize a semi-norm for the Alexander polynomial of a connected, compact, oriented 3-manifold on its first cohomology group to a semi-norm for an arbitrary Laurent polynomial f on the dual vector space to the space of exponents of f.…

In the present paper, we introduce Eulerian polynomials with a and b parameters and give the definition of them. By using the definition of generating function for our polynomials, we derive some new identities in Theory of Analytic…

The exponential generating functions of {n^(n+m)} for arbitrary integer m are expressed as rational functions of the e.g.f. of {n^(n-1)} [the tree function] and then of the e.g.f. of {n^n} [the endofunction function]. The coefficients in…

The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian…

We obtain two sequences of rational numbers which converge to the Euler-Gompertz constant. Denote by <f(x)> the integral of f(x)e^{-x} from 0 to infinity. Recall that the Euler-Gompertz constant \delta is <ln(x+1)>. Main idea. Let P_n(x) be…

The aim of this paper is to study degenerate Eulerian polynomials and degenerate Eulerian numbers, respectively as degenerate versions of the Eulerian polynomials and the Eulerian numbers, and to derive some of their properties.…

The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuel's multiplicities. In particular, we can give a purely algebraic proof of Bernstein's…

A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of…

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…

The central binomial series at negative integers are expressed as a linear combination of values of certain two polynomials. We show that one of the polynomials is a special value of the bivariate Eulerian polynomial and the other…

It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by…

We give a new identity involving Bernoulli polynomials and combinatorial numbers. This provides, in particular, a Faulhaber-like formula for sums of the form $1^m (n-1)^m + 2^m (n-2)^m + \cdots + (n-1)^m 1^m$ for positive integers $m$ and…

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials…

The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra $\mathfrak{g}$, the Eulerian idempotent is a canonical projection from the enveloping algebra $U(\mathfrak{g})$ to $\mathfrak{g}$.…

Let $\Phi$ be a root system. Postnikov introduced and studied the mixed $\Phi$-Eulerian numbers. These numbers indicate the mixed volumes of $\Phi$-hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian…

This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the…

In this paper we consider mixed volumes of combinations of hypersimplices. These numbers, called "mixed Eulerian numbers", were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan…